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Digital Signal Processing 23 (2013) 390–400
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Digital Signal Processing
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Multiple fundamental frequency estimation based on sparse representations in
a structured dictionary ✩,✩✩
Michal Genussov, Israel Cohen ∗
Department of Electrical Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel
a r t i c l e
i n f o
Article history:
Available online 11 September 2012
Keywords:
Piano transcription
Music information retrieval
Sparse representations
Multi-pitch estimation
a b s t r a c t
Automatic transcription of polyphonic music is an important task in audio signal processing, which
involves identifying the fundamental frequencies (pitches) of several notes played at a time. Its difficulty
stems from the fact that harmonics of different notes tend to overlap, especially in western music. This
causes a problem in assigning the harmonics to their true fundamental frequencies, and in deducing
spectra of several notes from their sum. We present here a multi-pitch estimation algorithm based on
sparse representations in a structured dictionary, suitable for the spectra of music signals. In the vectors
of this dictionary, most of the elements are forced to be zero except the elements that represent the
fundamental frequencies and their harmonics. Thanks to the structured dictionary, the algorithm does
not require a diverse or a large dataset for training and is computationally more efficient than alternative
methods. The performance of the proposed structured dictionary transcription system is empirically
examined, and its advantage is demonstrated compared to alternative dictionary learning methods.
© 2012 Elsevier Inc. All rights reserved.
1. Introduction
Transcription of music is defined as the process of identifying
the parameters of an acoustic musical signal, which are required
in order to write down the score sheet of the notes [1]. One of the
most important parameters is the pitch, which is represented in
written music by the note symbol. For convenience, we shall refer
here to the task of pitch identification as “transcription”. Automatic
transcription of music is important, since it allows structured audio coding, it is a helpful tool for musicians, enables modifying,
rearranging and processing music in an efficient way, and is the
basis for interactive music systems.
We need to separate between two different cases – transcription of monophonic music and transcription of polyphonic music.
Monophonic music is the case in which a single note is played at
each time instant. For this case, automatic transcription is practically a solved problem. Several proposed algorithms are reliable,
commercially applicable and operate in real time. However, transcription of polyphonic music, in which more than one note is
played at a time, is much more complicated. To the best of our
✩
This research is part of the M.Sc. thesis of the first author, M. Genussov, “Transcription and classification of audio data by sparse representations and geometric
methods”, Technion, Israel Institute of Technology, 2010.
✩✩
This research was supported by the Israel Science Foundation (grant no. 1130/
11).
Corresponding author.
E-mail addresses: [email protected] (M. Genussov),
[email protected] (I. Cohen).
*
1051-2004/$ – see front matter © 2012 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.dsp.2012.08.012
knowledge, no existing algorithm can identify multiple pitches in
an accuracy close to 100%. This is somehow not intuitive to understand, since when a trained human (such as a musician) listens to
a polyphonic music piece, he can distinguish between the different
notes and identify them, although played simultaneously. The difficulty stems from the fact that most often, especially in western
music, several harmonics of different notes overlap. This causes a
problem in assigning the harmonics to their true fundamental frequencies, and in deducing spectra of several notes from their sum
[2,3].
Since the 1970s, when Moorer built a system for transcribing
duets [4], there has been a growing interest in transcribing polyphonic music. The offered algorithms can be divided into three
main groups: time-based, frequency-based and time–frequencybased algorithms. The time-based group includes methods which
are based on the autocorrelation function [5–7] and on the periodicity of the signal [8–10]. The frequency-based group includes
methods which are based on typical harmonic patterns in the frequency domain [11,12], and which can be mapped to a logarithmic
scale to better fit the human auditory system [13–16]. The combined time–frequency-based group includes methods which use a
time–frequency image, such as a spectrogram or a scalogram [17],
or cepstrum analysis [18,19].
Some of the above methods [11,19] are based on the assumption that the spectral shape of the harmonics can be modeled by
a constant function, therefore spectrums of several notes can be
deduced from their combination. This is very inaccurate, since the
spectral shape changes as a function of many factors, which include the type of the musical instrument, the total intensity of the
M. Genussov, I. Cohen / Digital Signal Processing 23 (2013) 390–400
tone, the fundamental frequency and the stage in the time envelope of the sound, i.e. the ADSR (attack, decay, sustain, release)
envelope of the note [20]. Other methods [10,12,15,21] are limited
by the fact that they are supervised methods, i.e., require a training
set of pieces in order to transcribe another musical piece.
The idea of using Sparse representations as a time-based or
frequency-based method for transcription of polyphonic music, was first suggested by Abdallah and Plumbley [22]. It was
later improved and expanded [8,23,24], and inspired other works
[25–27].
The term “Sparse representations” means writing a signal as
a linear combination of very few underlying functions, which are
contained in a dictionary of underlying functions. This is implemented by multiplying the dictionary of the underlying functions
by a sparse vector (i.e., a vector that contains very few nonnegative elements compared to its length), giving the method its
name.
The motivation for applying sparse representations of music
signals, is that in played music, only a small number of notes
is played simultaneously compared to the number of available
notes. In the frequency domain, the approach is based on the idea
that power spectra of different notes approximately add, assuming random phase relationships. However, the existing algorithm
for transcription of music using sparse representations, developed
by Abdallah and Plumbley [24], suffers from a major drawback – it
requires a large and diverse database of notes, in order to build a
representative and reliable dictionary for the transcription. For example, if a note is played only in combination with other notes,
the learned dictionary will not represent the individual note, but
the combination in which it resides.
Several researchers, such as Raczynski et al. [28], Hennequin
et al. [29], Ewert and Müller [30] and Vincent et al. [31–33],
used Non-negative Matrix Factorization (NMF) for transcription
of polyphonic music. Some of their work involved representing
the signal using an adaptive structured decomposition, which is
quite similar to our approach. However, while they used NMF for
the decomposition, we used sparse representation and structured
dictionary learning, which involve different updating of the decomposed parameters. A more detailed comparison is introduced
in the Conclusions.
In this paper, we present an algorithm for multiple pitch estimation based on sparse representations in a structured dictionary,
suitable for the spectra of music signals. It achieves good estimation results even with a small and limited database of notes, thus
overcoming the drawback of the former algorithm based on sparse
representations. We show the advantages of our algorithm over
other algorithms for transcription of polyphonic music, using experiments on synthesized and recorded piano music.
The structure of the paper is as follows: In Section 2, we describe the musically-structured algorithm, based on sparse representations and spectral analysis of music signals. In Section 3,
we review the implementation of the algorithm in a transcription
setup for polyphonic music. In Section 4, experimental results are
presented, and the algorithm is compared to other transcription
methods. In Section 5, we conclude the paper and suggest future
expansions.
2. Musically-Structured (MS) sparse representations
Our transcription algorithm is based on sparse representations
with our new parametric dictionary, suitable for the spectra of music signals. We define this dictionary as the Musically-Structured
dictionary. The algorithm is composed of two stages: Sparse coding
and structured dictionary learning.
391
2.1. Sparse coding
The purpose of sparse coding is to approximate the solution of
the following ( P 0 ) problem:
( P 0 ):
min x0
x
subject to
Ax = y.
(1)
The vector x is a sparse vector, encoding the columns (atoms)
of the dictionary A, such that together they represent the signal y.
This method was shown useful to capture the meaningful characteristics of signals, such as images and audio [34].
An error-tolerant modified version for multiple signals:
ε
P0 :
min Y − AX2F
X
s.t.
xi 0 < K , 1 i M ,
(2)
where F is the Frobenius norm, and xi are the columns of X. In our
case, each of the columns of the matrix Y, i.e., the signals, is the
Constant Q Transform (CQT) [35], at a certain time window. We
use the CQT instead of the Short-Time Fourier Transform (STFT),
since it is more suitable for the auditory system. The STFT can be
modeled as a bank of linearly spaced filters, whose bandwidths
are constant. On the other hand, the CQT can be modeled as a
bank of logarithmically spaced filters, whose bandwidths are logarithmically increasing as a function of frequency. The Q parameter,
which is the bandwidth to center–frequency ratio, is constant as a
function of the frequency. In the original CQT transform, the length
of the time window decreases as a function of frequency. We use
a degenerated version of it, with a constant window length. Because of its logarithmic frequency scale, The CQT better fits the
auditory system than the STFT [35], and also reduces the computational complexity by using less frequency bins than the STFT.
The matrix A ∈ Rn×m is the Musically-Structured dictionary in
which each atom (column) is the CQT of a single note. Each column in X ∈ Rm× M encodes a linear combination of the notes from
A which are played in the time window of the corresponding column in Y ∈ Rn× M . At each time window, the corresponding column
in X is determined as the sparsest column which minimizes the
Frobenius norm Y − AX F . In our algorithm we use a greedy
method for sparse coding, since it allows to pre-define the cardinality (the number of the non-zero elements K ) of the sparse
vector, according to the evaluated number of notes at each time
window. Specifically, we choose to use the OMP algorithm [36,37],
which settles a good compromise between complexity and performance, compared to other algorithms for sparse coding, such as
Matching Pursuit (MP) or Basis Pursuit (BP) algorithms [34].
2.2. Musically-Structured dictionary
The dictionary matrix A can be chosen in three different manners:
(1) Analytic dictionary – a predefined dictionary, which is the inverse (or pseudo-inverse) matrix of a transform such as the
Curvelet transform [38,39], the Contourlet transform [40,41]
the short-time Fourier transform or the Wavelets transform
[42].
(2) Learned (explicit) dictionary – a dictionary is learned blindly
from a set of signals or samples yi ,{1i M } , in an iterative and
alternative manner, arranged as columns of a matrix Y. The
optimization problem ( P 0ε ) turns into:
min Y − AX2F
A,X
s.t.
xi 0 < K , 1 i M .
(3)
Abdallah and Plumbley [22,24] used such a learned dictionary
for transcription of polyphonic music.
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M. Genussov, I. Cohen / Digital Signal Processing 23 (2013) 390–400
Fig. 1. Short-time Fourier transform of an A4 piano note.
(3) Parametric dictionary – a compromise between a pre-defined
analytic dictionary to a data-driven learned dictionary. Only
a limited set of parameters in the dictionary is learned, and
the rest is pre-defined. It represents the data better than an
analytic dictionary, and is computationally more efficient than
an explicit learned dictionary. In addition, it can avoid overfitting if built wisely, unlike an explicit learned dictionary.
Our motivation is to develop a parametric dictionary which is
suitable for the spectrum of music signals. In order to do so, we
use the common features in the spectrum of a musical note. The
magnitude of the Fourier transform of every note can be modeled by an impulse train at the fundamental frequency and at its
harmonics, which has been multiplied with a shaping filter, that
is controlled by the factors mentioned in Section 1 – the musical instrument, the intensity, the duration of the note and the
fundamental frequency. For illustration, see the short-time Fourier
transform (STFT) of an A4 piano note in Fig. 1. This STFT illustration was performed on a note played over a long time, so we could
use a large time window, and get good resolution both in time and
in frequency.
If we apply the Constant Q transform instead of the STFT on
a A4 piano note, we get peaks of the fundamental frequency and
its harmonics in intervals which become logarithmically smaller as
the CQT bin grows (see bottom of Fig. 2). The dictionary which
we offer is initialized such that each atom is the modeled CQT
of a different note in the piano (total – 88 notes). More specifically, we initialize each atom by an impulse train of 6 elements,
corresponding to the suitable fundamental frequency and its first
5 harmonics. This number of harmonics was found to be sufficient in multiple tests and on a wide range of notes. We multiply
this finite impulse train by an initial shaping function f (n) = n1 ,
where n is the partial number. Finally, we map the dictionary to
the CQT scale. We denote this dictionary as the initial MusicallyStructured dictionary. An image of this initial dictionary, and the
CQT of a note represented by a certain atom in it, are presented in
Fig. 2.
The next step is learning certain parameters in the MusicallyStructured dictionary. The support of the dictionary is constant, i.e.,
the location of the non-zero elements in the vector, which are the
CQT bins of the fundamental frequencies and their 5 first harmonics. However, the entries of the elements in the support are
learned, i.e., the amplitudes of the fundamental frequencies and
their harmonics. These amplitudes are an important feature of the
timbre of the signal, which was mentioned before. The minimal
resolution required for music transcription is 12 CQT frequency
bins per octave (one for each 100 cent = semitone). However, using
a higher resolution improves the transcription results for poly-
Fig. 2. The initial MS dictionary (top) and the CQT of an A4 note represented by a
certain atom (bottom). The atom which represents the note in the bottom picture
is marked by a rectangle in the dictionary.
phonic music. In the experiments presented in Section 4 we show
the results for a mapping to 24 frequency bins per octave (one for
each 50 cents = quarter tone).
The MS-dictionary is expected to have several advantages over
the explicit learned dictionary proposed by Abdallah and Plumbley [24], as well as advantages over an analytic and pre-defined
appropriately dictionary:
(1) Avoids overfitting – each atom in the MS dictionary represents
a single note, even if it does not appear individually in the
dataset (e.g., it is played only as part of a chord). We expect
this requirement to be fulfilled, since the support of the dictionary is constant, and since each atom in the initial dictionary
represents an individual note.
(2) Better representation of the signal than an analytic dictionary
– this is due to the fact that the entries in the support of the
dictionary matrix are learned according to the timbre of the
signal.
(3) Reduced complexity – the complexity of learning the dictionary is proportional to the number of the learned parameters.
In an explicit dictionary, the number of learned parameters
equals the size of the dictionary, i.e., nm. In the MS-dictionary,
M. Genussov, I. Cohen / Digital Signal Processing 23 (2013) 390–400
the number of learned parameters is hm, where h is the number of partials (fundamental frequency + harmonics), which we
choose to be 6. Since h < n, the complexity of the dictionary
is reduced.
The formulation of the new problem is:
( P MS ):
min Y − AX2F
A,X
s.t.
xi 0 K ∀i ∈ {1, . . . , m}
a j P cj = 0 ∀ j ∈ {1, . . . , M },
(4)
where P cj is the subset of indices out of the support of the atom a j .
In order to approximate the solution to P MS , we apply a modified
MOD (Method-Of-Directions) or K-SVD dictionary learning algorithm according to the MS parametric dictionary.
The MS-dictionary is composed of 88 atoms, where each atom
represents a different note. Musical pieces usually do not contain
all of the 88 notes. Therefore, in the dictionary learning stage,
we update only the atoms which were used for the sparse coding stage. We denote the dictionary matrix of the used atoms in
A(k) , and the corresponding coefficients matrix
the kth iteration as as X(k) . The rest of the atoms remain unchanged, and are added to
the dictionary after the update. In the following, we describe the
modified MOD and K-SVD dictionary learning algorithms.
Table 1
The Musically-Structured MOD algorithm.
Task: Train a dictionary A to sparsely represent the log spectrogram matrix Y,
by approximating the solution to Problem (4).
Initialization: Initialize k = 0, and
• Initialize Dictionary: Build A(0) ∈ Rn×m , with non-zero entries only in the
locations corresponding to the fundamental frequencies and their
harmonics.
• CQT mapping: Map the dictionary to the CQT logarithmic scale.
• Normalization: Normalize the columns of A(0) to unit 2 -norm.
Main iteration: Increment k by 1, and apply
• Sparse Coding Stage: Use a pursuit algorithm to approximate the solution
of
xi = argminx y i − A(k−1) x22 subject to x0 K
obtaining sparse representations xi for 1 i M. These form the matrix
X(k) .
• Define the used atoms: The rows in X(k) which have an 1 -norm above
a certain threshold correspond to the used atoms in the dictionary matrix.
X(k) and the matrix composed
These rows are concatenated and defined as of the used atoms is defined as A.
• MOD Dictionary Update Stage: Update the dictionary of used atoms A(k)
by the matrix X(k) :
†
A(k) = argminà Y − A
X(k) 2F = Y
X(k) .
• Zero the entries out of the support: Zero the entries out of the support
which was defined in the initial dictionary.
• Add the unused atoms: Add the unused atoms to A(k) .
This is the updated dictionary A(k) .
• Stopping Rule: If the change in Y − A(k) X(k) 2F is small enough, stop.
Otherwise, apply another iteration.
Output: The desired outputs are A(k) and X(k) .
2.3. Musically-Structured MOD
In the MOD algorithm, developed by Engan et al. [43], the dictionary update stage is conducted using least squares:
A(k) = argmin Y − AX(k) 2F = YX† ,
Y − AX2F
†
A(k) = argmin Y −
A
X(k) 2F = Y
X(k) .
A
2
m
T
= Y −
a jx j j =1
(5)
j = j 0
(6)
The matrix X is composed of the concatenation of the rows of X
which have a 1 -norm above a certain threshold, and correspond
to the used atoms in A. This diminution is intended to prevent
singularity of X† , and deals with (most of) the cases, where the
musical piece does not contain all of the 88 notes.
The normalization stage of the atoms in the MOD (and in the
K-SVD) is intended to make the implementation of the sparse coding stage simpler when using a greedy method, and it does not
change its solution.
After updating the atoms, we zero the elements out of the original support. The MS-MOD is summarized in Table 1. The main
modifications compared to the original MOD are indicated by underlined text.
2.4. Musically-Structured K-SVD
A different update rule for the dictionary was proposed by
Aharon et al. [44], leading to the K-SVD algorithm. In this algorithm, the atoms (i.e., columns) in the dictionary A are handled
sequentially. The dependency on the atom a j 0 in (3) is isolated by
rewriting the term Y − AX2F :
F
2
T
T −
=
Y
−
a
x
a
x
.
j
j
0 j0 j
A
where X† is the Moore–Penrose pseudo-inverse of X. After an initialization of the dictionary matrix, the matrix of sparse columns
X and the dictionary matrix A are updated alternately at each iteration, until the change at the kth iteration of Y − A(k) X(k) 2F is
small enough.
In our modified MOD algorithm, the MS-MOD, the dictionary
matrix is updated as in the MOD algorithm:
393
(7)
F
In this description x Tj stands for the jth row of X, i.e., the coefficients which correspond to the jth atom. We define the term
E j0 = Y −
a j x Tj
(8)
j = j 0
as the error matrix corresponding to the atom a j 0 . We restrict it
only to the columns that correspond to the signals (columns) in
Y which use the atom a j 0 , and denote the restricted error matrix
as E Rj . Both a j 0 and the non-zero elements in x Tj , which are de0
0
noted by (x Rj ) T , are updated in this algorithm, by minimizing the
0
term in (7), using a rank-1 approximation of the error matrix E Rj .
0
This approximation is obtained via singular value decomposition
(SVD).
In our modified algorithm, the MS-KSVD, we update only the
elements in the support, for each atom individually. For each
atom a j 0 , the error matrix E j 0 is defined as in the K-SVD. Its
columns are restricted as in the K-SVD, but now its rows are also
restricted, according to the support of the atom a j 0 . Thus, updating only the support of a j 0 , we denote this restricted error matrix
E Rj and the elements in the support of a j 0 as a j 0 . The vectors
as 0
a j 0 and (x Rj ) T are updated using rank-1 approximation of the er0
E Rj by singular value decomposition. The algorithm is
ror matrix 0
summarized in Table 2. The main modifications compared to the
original K-SVD are indicated by underlined text.
3. Implementation
The overall transcription algorithm which we offer is as follows:
(1) Note onset detection – since we focus only on pitch identification, we conduct this stage manually, or extract the onsets
394
M. Genussov, I. Cohen / Digital Signal Processing 23 (2013) 390–400
4. Experiments
Table 2
The Musically-Structured K-SVD algorithm.
Task: Train a dictionary A to sparsely represent the log spectrogram matrix Y,
by approximating the solution to Problem (4).
Initialization: Initialize k = 0, and
• Initialize Dictionary: Build A(0) ∈ Rn×m , with non-zero entries only in the
locations corresponding to the fundamental frequencies and their
harmonics.
• CQT mapping: Map the dictionary to a the CQT logarithmic scale.
• Normalization: Normalize the columns of A(0) to unit 2 -norm.
Main iteration: Increment k by 1, and apply
• Sparse Coding Stage: Use a pursuit algorithm to approximate the solution
of
i = argminx yi − A(k−1) x22 subject to x0 K
x
i for 1 i M. These form the matrix
obtaining sparse representations x
X(k) .
• KSVD Dictionary Update Stage: Update the support of each atom
a j 0 ,{ j 0 =1,2,...,m} in the dictionary matrix by rank-1 approximation of its
error matrix E Rj , using SVD.
0
E R is the restriction of the matrix E j = Y −
a j x T to the columns
j0
0
j = j 0
j
that correspond to the samples in Y which use the atom a j 0 ,
and to the rows that correspond to the support of a j 0 .
After applying the SVD E Rj = UVT , update the support of the dictionary
0
atom by a j 0 = u1 and their representation coefficients
R T
T
by (x j ) = ([1, 1]v1 ) .
0
• Add the unused atoms: Add the unused atoms to A(k) .
This is the updated dictionary A(k) .
• Stopping Rule: If the change in Y − A(k) X(k) 2F is small enough, stop.
Otherwise, apply another iteration.
Output: The desired outputs are A(k) and X(k) .
from a MIDI file in the case of transcribing a synthesized MIDI
musical piece.
(2) Evaluation of the number of notes – performed 32 ms after the
onset, in a 64 ms time window, either manually or extracted
from a MIDI file (from the same reason as in the previous
stage). This number is defined as K , it is given as input to
our algorithm at each relevant time frame, and it is used as
the maximal cardinality of the sparse vector during the transcription process.
(3) Constant Q transform is applied on the signal in the 64 ms
time window.
(4) All the vectors of CQTs of the time windows mentioned before
are concatenated in columns of the signal matrix Y (which is
actually a log-spectrogram), and a Musically-Structured dictionary learning algorithm is applied on the matrix Y to transcribe the music in each of the time windows represented by
its columns.
The reason for applying the multi-pitch estimation only on
64 ms time windows, 32 ms after the onsets of the notes, is that
the acoustic spectrum of a tone significantly changes as a function
of the stage in the ADSR envelope. We wish to sample all notes
at the same stage, such that the atoms in the dictionary would
represent them well. We assume that after 32 ms, the ADSR envelope is in its sustained stage, which is the most suitable stage
for the spectral analysis because of its stability and relatively long
duration. A block diagram of the overall algorithm is presented in
Fig. 3.
4.1. Experimental setup
We perform pitch identification of synthesized piano music
from MIDI files, and pitch identification of real recorded piano music. We do not deal with finding the onset and offset instances
of the notes, neither with finding the number of notes played at
each time (which we use as the maximal number of notes – K ,
for the sparse coding). The stopping criterion for the algorithms
is achieved when there is a change of less than 5% in the Frobenius norm of the residual. We compare the performance of the
MS-MOD and MS-K-SVD to that of the standard MOD and K-SVD
algorithms (which use an explicit, blindly learned dictionary), and
to that of an analytic dictionary (un-learned) with OMP in the
sparse coding stage. The analytic dictionary is determined as the
initial MS-dictionary. In the standard MOD and K-SVD we decide
on the fundamental frequency of individual atoms that are learned,
by a method called HPS (Harmonic Product Spectrum), [45] which
is intended for pitch estimation in monophonic music.
We use different measures for the evaluation of the results: The
first is the Accuracy measure, defined by [46]
Accuracy =
TP
TP + FP + FN
(9)
.
The term TP is the number of true positives (correct detections),
FP is the number of false positives and FN is the number of false
negatives. When Accuracy = 1, it means that all the notes are identified correctly and there are no false positives nor false negatives.
The second measure is the transcription error score [12]. If we denote by N sys the number of reported pitches, by N ref the number
of ground-truth pitches and by N corr their intersection, then the
transcription error score across all time frames t is:
T
E tot =
t =1 max( N ref (t ), N sys (t )) −
T
t =1 N ref (t )
N corr (t )
.
The MIDI files for the experiments include a monophonic musical piece, a simple polyphonic piece (mainly two or three notes
played at a time, a limited variety of notes), a complicated polyphonic piece (mainly five or six notes played at a time, a large
variety of notes) and a piece of chords. Their piano rolls are presented in the next subsection. These MIDI files are synthesized
with a sampling frequency of 44.1 kHz, by FM-synthesis, using the
“Matlab and MIDI” software [47]. The MS-K-SVD code is a modified
version of the K-SVD code [48]. The recorded piano files include a
monophonic piece and a piece of chords. These pieces are recorded
on a Yamaha U1 piano, and saved with a sampling frequency of
44.1 kHz.
We also compare our transcription method to former reported
transcription results (Costantini et al. [15], Poliner and Ellis [12],
Ryynänen and Klapuri [21] and Marolt [10]), which were examined on a set of polyphonic classical synthesized MIDI music pieces
which were collected from the Classical Piano Midi Page [49]. The
list of 130 pieces set is specified in [12]. The first minute from each
song was taken. The 130 pieces set was randomly split into 92
training, 24 testing and 13 validation pieces (we used for our tests
only the testing set since no training is needed). In addition to the
Fig. 3. A block diagram of the overall transcription algorithm.
M. Genussov, I. Cohen / Digital Signal Processing 23 (2013) 390–400
synthesized audio, piano recordings were made from a subset of
the classical MIDI files using a Yamaha Disklavier playback grand
piano. Twenty training files and ten testing files were randomly
selected for recording. The recorded files are available at [50].
4.2. Results – synthesized MIDI music
First we present the performance of the MS-dictionary learning
algorithm on a monophonic piece, a simple polyphonic piece and
a complicated polyphonic piece. The transcription results are presented in Tables 3 and 4. The corresponding piano rolls and their
identification by MS-MOD and MS-KSVD are presented in Figs. 4,
5 and 6. From the tables we can learn that:
(1) All methods perfectly identify the notes in the monophonic
music.
Table 3
Transcription Accuracy percentage for three different types of songs.
MS-MOD
MS-K-SVD
MOD
K-SVD
Analytic dictionary
Monophonic
music
Simple polyphonic
music
Complicated
polyphonic music
100
100
100
100
100
69.6
67.7
39.5
37.6
45.8
64.0
64.5
43.5
42.7
41.0
Table 4
Transcription E tot percentage for three different types of songs.
MS-MOD
MS-K-SVD
MOD
K-SVD
Analytic dictionary
Monophonic
music
Simple polyphonic
music
Complicated
polyphonic music
0
0
0
0
0
17.9
19.3
44.5
46.2
37.2
23.6
23.2
41.0
41.7
43.3
395
(2) The estimation performance of the algorithms based on the
parametric MS dictionary is better than that of the explicit dictionaries as well as the analytic dictionaries when identifying
notes in polyphonic music. The advantage is more significant
in the simple polyphonic music than in the complicated polyphonic music (where the dataset of notes is larger and richer).
(3) The estimation performance of the MS-dictionary and the analytic dictionary become worse when the polyphonic music
becomes more complicated. On the contrary, the performances
of the methods based on explicit dictionaries are improved.
This implies that the explicit dictionaries need a large data-set
in order to achieve good performance.
From the piano rolls we can identify some mistakes in the transcription as spurious notes (mistakes of semitones), which might
be fixed by using a higher frequency resolution. Some mistakes are
associated with notes that share a similar spectral shape, such as
notes whose difference is an integer number of octaves, or notes
that share common harmonics with the true note.
We now turn to the problem mentioned in Section 1 – deduction of notes from chords. This problem is hard, since the notes in
a chord share multiple harmonics. We compare the results of the
MS-dictionary to that of an explicit dictionary, and of an analytic
dictionary. The piano rolls of the original and identified music are
presented in Fig. 7. In the case of the chords, MS-MOD, MS-K-SVD
and OMP with an analytic dictionary identify all the notes, despite
of the difficulty of this task. The MOD and K-SVD identify only the
lower notes.
We compare the former reported results, obtained by the implementations of original authors (Costantini et al. [15], Poliner
and Ellis [12], Ryynänen and Klapuri [21] and Marolt [10]), on the
set of polyphonic classical music, which was described in the beginning of this section, to the multi-pitch estimation of the testing
set by MS-MOD, MS-K-SVD, MOD, K-SVD and OMP with an ana-
Fig. 4. The ground-truth piano roll (left) and the identified piano roll of a monophonic piece using MS-MOD (middle) and using MS-K-SVD (right). All the notes were identified
correctly.
Fig. 5. The ground-truth piano roll (left) and the identified piano roll of a simple polyphonic piece using MS-MOD (middle) and using MS-K-SVD (right). Black = true positive,
red = false positive, yellow = false negative. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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M. Genussov, I. Cohen / Digital Signal Processing 23 (2013) 390–400
Fig. 6. The ground-truth piano roll (left) and the identified piano roll of a complicated polyphonic piece using MS-MOD (middle) and using MS-K-SVD (right). Black = true
positive, red = false positive, yellow = false negative. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this
article.)
Fig. 7. The ground-truth piano roll and the identified piano roll of synthesized chords using different methods. Black = true positive, red = false positive, yellow = false
negative. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
lytic dictionary. However, the results should be compared carefully
due to the following differences between the algorithms:
(1) The former transcription algorithms are supervised methods,
i.e. they are based on the training set mentioned before. Their
results presented here are after training on pieces which were
written by the same composers as in the testing set. Our algorithm, and the other transcription methods based on sparse
coding, are unsupervised methods, and they are tested on the
same testing set without training.
(2) The number of notes in each frame was inserted as a parameter to the pitch estimation methods based on sparse representations (MS-MOD, MS-K-SVD, MOD, K-SVD and OMP with
an analytic dictionary), as the maximal cardinality K of each
vector of coefficients x. However, the former transcription algorithms did not receive the maximal cardinality as a parameter.
A comparison of the results on synthesized polyphonic classical music are presented in Table 5. From the table one can see
Table 5
Transcription performance on synthesized polyphonic classical music.
MS-MOD
MS-K-SVD
MOD
K-SVD
Analytic dictionary
Costantini et al.
Poliner and Ellis
Accuracy (%)
E tot (%)
58.7
59.8
39.4
31.0
39.1
72.3
64.7
28.2
27.2
45.3
54.1
45.1
20.1
41.7
that the results of the transcription by MS-dictionary learning algorithms outperform those of the other unsupervised methods for
transcription using sparse representations (MOD, K-SVD and OMP
with an analytic dictionary). They are inferior compared to those
of other transcription methods, but, as mentioned before, not directly comparable. The measures Accuracy and E tot are not always
correlated in their performance, due to their different definitions.
Therefore, they are both used for evaluation.
M. Genussov, I. Cohen / Digital Signal Processing 23 (2013) 390–400
397
Fig. 8. The ground-truth piano roll and the identified piano roll of recorded monophonic piano music using different transcription methods. All the notes were identified
correctly.
Fig. 9. The ground-truth piano roll and the identified piano roll of recorded piano chords using different transcription methods. Black = true positive, red = false positive,
yellow = false negative. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
4.3. Results – recorded piano music
The task of multiple pitch estimation of real recorded piano
music is much harder, since it entails several additional obstacles, such as beginning transients of notes, a noisy environment,
formants produced by the shape of the musical instrument, reverberations of the room and strings inharmonicity. In a real vibrating
string, the harmonics are not located exactly in multiple integers of
the fundamental frequency, and the musical instruments are tuned
accordingly [51].
First we perform some simple tests, on monophonic music and
on chords. We compare the performances of MS-MOD and MSK-SVD to that of MOD, K-SVD, and OMP with an analytic dictionary, as presented in Figs. 8 and 9. All the methods perfectly iden-
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M. Genussov, I. Cohen / Digital Signal Processing 23 (2013) 390–400
Table 6
Transcription performance on recorded polyphonic classical music.
MS-MOD
MS-K-SVD
MOD
K-SVD
Analytic dictionary
Costantini et al.
Poliner and Ellis
Accuracy (%)
E tot (%)
54.3
54.3
22.9
19.8
43.9
59.2
56.5
30.9
31.1
63.3
67.4
40.1
33.3
46.7
Table 7
Transcription performance on synthesized plus recorded polyphonic classical music.
MS-MOD
MS-K-SVD
MOD
K-SVD
Analytic dictionary
Costantini et al.
Poliner and Ellis
Ryynänen and Klapuri
Marolt
Accuracy (%)
E tot (%)
57.4
58.2
34.5
27.7
40.5
68.0
62.3
56.8
30.4
29.0
28.3
50.6
58.0
43.6
24.6
43.2
46.0
87.5
tify the notes in the monophonic music. The detection of chords is
worse than in synthesized music, and this may stem from the obstacles mentioned before. Still, the performances of the MS-MOD,
MS-K-SVD and OMP with an analytic dictionary are better than
those of the MOD and K-SVD, which identify one note at a time
while there are actually two or three notes.
We compare the transcription results on the recorded classical
polyphonic music pieces to those of previous works. The results of
transcription on recorded polyphonic music are presented in Table 6, and the results of transcription on both synthesized and
recorded polyphonic music are presented in Table 7. In recorded
classical piano music, similarly to synthesized classical piano music, the results of the transcription by MS-dictionary learning algorithms outperform those of the other unsupervised methods for
transcription using sparse representations (MOD, K-SVD and OMP
with an analytic dictionary). In this case they have similar results to other transcription methods, and even outperform some
of them, although they are supervised methods.
5. Conclusions
We have presented a multiple fundamental frequency estimation system based on sparse representations. The power spectrum
(CQT) of the music signal at each time window is represented as a
multiplication of a learned dictionary with a sparse vector. This
framework relies on the assumptions that the number of notes
played at a time is small compared to the number of available
notes, and that the power spectrum of different notes approximately add, for random phase relationships. We offered a parametric dictionary, namely “Musically-Structured dictionary”, based on
the common features of the power spectra of music signals. This
parametric dictionary is more suitable for multiple pitch estimation than an analytic dictionary or an explicit dictionary. We developed modifications of two dictionary learning algorithms, MOD
and K-SVD, which are denoted MS-MOD and MS-K-SVD, respectively, for learning the parametric dictionary.
The performance of the MS-dictionary was examined empirically on MIDI synthesized music and on recorded music. In the
polyphonic pieces we examined, most of the notes were recognized correctly. Relatively good performance was seen also in cases
of small datasets with overlapping harmonics, such as chords or
octaves. It is shown that the transcription using MS-dictionary outperforms transcription using an analytic or an explicit dictionary.
The advantage over an explicit dictionary grows as the dataset is
smaller, and as there are more overlapping harmonics.
The proposed algorithm is advantageous from several reasons:
It adapts to the timbre of the signal and reduces computational
complexity compared to sparse representations with analytic dictionaries. It avoids overfitting to small datasets or to notes played
conjugately (such as chords), thus outperforming sparse representations with explicit dictionaries. In addition, it is as unsupervised
method, thus a training set is not required. Our framework can
be modified and expanded in several ways: The dictionary can
be better adapted to real recorded music. The overtones of real
strings are not exactly multiple integers of the fundamental frequency (string inharmonicity). Therefore, a dictionary in which the
non-zero elements are non-uniformly spaced, might lead to better
transcription results. Another possible expansion is to add more
atoms to the dictionary. The timbre of the sound changes as a
function of its intensity. Therefore, it is reasonable that each note
would be represented by several atoms, where each atom represents a different intensity. Each group of atoms which represents
the same note would have the same support, but the values of the
non-zero elements would be changed individually, such that each
atom would fit another time window (or group of time windows)
within the note.
Another possibility is to exploit high-level information. Former works [9,12] used prior information on the structure, tempo,
rhythm and key of the musical piece, as well as expected relations
between consecutive notes. This can be added to our framework
as a subsequent stage, or as an iterative stage, for example via a
Hidded Markov Model (HMM).
As for the comparison to non-negative matrix factorization
(NMF) mentioned in Section 1, we would like to point out the similarities and dissimilarities to our work. Both approaches – NMF
and sparse representations, deal with the factorization problem
AX = Y. Moreover, sparse representations can be considered as a
special case of NMF, in which the sparseness constraint is imposed [34]. In addition, in the specific NMF algorithms mentioned
in Section 1, the initialization and update of the matrix A are based
on a special “musically-based” structure, similar to ours. However,
the major difference between the NMF approach and our approach
is in the update rule – while the NMF algorithms update both matrices A and X by minimizing the Bregman divergence between the
matrix Y and its approximation AX, we use pursuit algorithms for
estimating the matrix X, and dictionary learning algorithms for estimating the matrix A. These algorithms are addressed specifically
for approximating the sparse representation of the signal, with our
adaptation to the problem of musical multi-pitch estimation. An
interesting test would be to compare the multi-pitch estimation of
our algorithm to that of the relevant NMF algorithms, on the same
data set.
A next step would be to extend the proposed framework to
multi-pitch estimation for other musical instruments, as well as
for their combinations. It would be interesting to examine the difference in the performance of the algorithm between western and
non-western instruments (e.g., [52]).
Acknowledgments
The authors thank Dr. Ron Rubinstein for fruitful discussion and
helpful suggestions. They also thank the anonymous reviewers for
their constructive comments and useful suggestions.
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Michal Genussov received the B.Sc. degree (Cum Laude) in Biomedical engineering in 2008, and the M.Sc. degree in Electrical engineering, in
the field of audio signal processing, in 2011, both from the Technion, Israel
Institute of Technology. Currently she works as an image processing algorithm engineer in Digital Optics Corporation, a company which develops
miniaturized cameras, mainly for the camera phones industry.
Israel Cohen is an Associate Professor of electrical engineering at the
Technion – Israel Institute of Technology, Haifa, Israel. He received the
B.Sc. (Summa Cum Laude), M.Sc. and Ph.D. degrees in Electrical Engineering from the Technion – Israel Institute of Technology, in 1990, 1993 and
1998, respectively.
From 1990 to 1998, he was a Research Scientist with RAFAEL Research
Laboratories, Haifa, Israel Ministry of Defense. From 1998 to 2001, he was
a Postdoctoral Research Associate with the Computer Science Department,
Yale University, New Haven, CT. In 2001 he joined the Electrical Engineering Department of the Technion. His research interests are statistical
signal processing, analysis and modeling of acoustic signals, speech enhancement, noise estimation, microphone arrays, source localization, blind
source separation, system identification and adaptive filtering. He is a
coeditor of the Multichannel Speech Processing section of the Springer
Handbook of Speech Processing (Springer, 2008), a coauthor of Noise Reduction in Speech Processing (Springer, 2009), a coeditor of Speech Processing in
Modern Communication: Challenges and Perspectives (Springer, 2010), and a
general co-chair of the 2010 International Workshop on Acoustic Echo and
Noise Control (IWAENC).
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M. Genussov, I. Cohen / Digital Signal Processing 23 (2013) 390–400
Dr. Cohen is a recipient of the Alexander Goldberg Prize for Excellence
in Research, and the Muriel and David Jacknow award for Excellence in
Teaching. He served as Associate Editor of the IEEE Transactions on Audio, Speech, and Language Processing and IEEE Signal Processing Letters,
and as Guest Editor of a special issue of the EURASIP Journal on Advances in
Signal Processing on Advances in Multimicrophone Speech Processing and
a special issue of the Elsevier Speech Communication Journal on Speech Enhancement.